Suppose $f$ is a classical modular form of weight $r$ for a (congruence) group $\Gamma$. Let $\gamma$ be any matrix in $\operatorname{SL}_2(\mathbb{Z})$. Then the slash operator $|_\gamma$ is usually defined as: $$f|_\gamma(z) = \frac{f(\gamma z)}{(cz+d)^r}$$ where $(c, d)$ stands for the bottom row of $\gamma$. The function $f|_\gamma$ is again a modular form of the same weight but for the conjugated group $\gamma^{-1} \Gamma \gamma$. I'm interested in some method to compute the first $10^5$ Fourier coefficients of $f|_\gamma$. The ones for $f$ are known. I'm specially interested in the case where $f$ is the cuspform associated to the elliptic curve $y^2 + yx + y = x^3 + 4x - 6$ (group $\Gamma_0(14)$).
I do not know if this is a simple task or not, as I barely know anything about computational algorithms for modular forms, but simply approximating the scalar product against an exponential function with good precision is probably a tricky approach.